American Journal of Theoretical and Applied Statistics 2017; 6(3): 156-160 http://www.sciencepublishinggroup.com/j/ajtas doi: 10.11648/j.ajtas.20170603.14 ISSN: 2326-8999 (Print); ISSN: 2326-9006 (Online)
On a New Class of Regular Doubly Stochastic Processes Reza Farhadian*, Nader Asadian Department of Mathematics and Statistics, Lorestan University, Khorramabad, Lorestan, Iran
Email address:
[email protected] (R. Farhadian),
[email protected] (N. Asadian) * Corresponding author
To cite this article: Reza Farhadian, Nader Asadian. On a New Class of Regular Doubly Stochastic Processes. American Journal of Theoretical and Applied Statistics. Vol. 6, No. 3, 2017, pp. 156-160. doi: 10.11648/j.ajtas.20170603.14 Received: March 17, 2017; Accepted: March 29, 2017; Published: May 25, 2017
Abstract: In this article, we show that the well-known Helmert matrix has strong relationship with stochastic matrices in modern probability theory. In fact, we show that we can construct some stochastic matrices by the Helmert matrix. Hence, we introduce a new class of regular and doubly stochastic matrices by use of the Helmert matrix and a special diagonal matrix that is defined in this article. Afterwards, we obtain the stationary distribution for this new class of stochastic matrices.
Keywords: Helmert Matrix, Stochastic Matrix, Markov Chain, Transition Probability, Stationary Distribution, Regular Chain, Ergodic Chain
1. Introduction In modern probability theory and dynamical systems, stochastic processes and Markov chains are applied contexts that are used in advanced sciences. Two basic topics in stochastic process are prediction and filtering. In the topic of prediction for Markov chains, we can obtain the -step transition probability, using the 1-step transition probability. This work is done by a matrix which is called the stochastic or probability or transition or Markov matrix. In stochastic processes or Markov chains, stochastic matrices are used for showing the transition probabilities [9, 11]. On the other hand, there is a special matrix in liner algebra that is called the Helmert matrix. A Helmert matrix of order is a square matrix that was introduced by H. O. Lancaster in 1965 [4]. Usually, the Helmert matrix is used in mathematical statistics for analysis of variance (ANOVA), see [1, 2, 8]. In this article, we will show that the Helmert matrix can be used in stochastic processes. For the next sections, the following notation will be used: (a) denotes an identity matrix of order . (b) denotes an × matrix whose elements are all 1. (c) denotes the inverse of a matrix . (d) denotes the transpose of a matrix . … (e) denotes an × diagonal matrix with diagonal entries , , …, . > 0 stands for a matrix all of whose elements (f) are positive.
(g) ℝ denotes real numbers.
2. Definitions and Particulars 2.1. Stochastic Matrix
Suppose that a stochastic process start from state to state . This transition shown by → , and denote its probability. Now, if process consist of states and denotes the state at time , then the transition → at time , is indicated by = and $ = . Furthermore, a process is called a Markov chain if the transition probability is independent of time for every states and of state space. Hence, the transition probability under the Markov property, is defined as: = %&'
$
= |
= %&'
= , $
= |
=
,…,
= )
=
)
(1)
for every , , … , , , of state space. Thus, we can construct a × stochastic (transition) matrix by the , 1≤ , ≤ : Definition 1. [9] A × real matrix such as % = , - × is called a stochastic matrix (or row stochastic matrix), if ≥ 0, 1 ≤ , ≤ 1)
American Journal of Theoretical and Applied Statistics 2017; 6(3): 156-160
2) ∑ 3 = 1, ∀ ∈ '1, 2, … , ) Definition 2. [5] A doubly stochastic matrix, is a square matrix of nonnegative real numbers with each row and column summing to 1(in other words, a stochastic matrix is doubly, if its transpose is stochastic matrix). Hence, If , - × be the 1-step transition matrix, then the
matrix , 456 - × is called the -step transition matrix (under the conditions of Definition 1). Chapman and Kolmogorov independently showed that if % is a 1-step transition matrix, then -step transition matrix denoted by % 456 and is equal to 78889888: × % × … × % % 456 = %5 = %
(2)
5 ;
